Groups with an automorphism of prime order
that is almost regular in the sense of
rank

Abstract

Let $\varphi$ be an automorphism of prime order $p$ of a finite group
$G$, and let $r$ be the (Pr\"ufer) rank of the fixed-point subgroup
$C_G(\varphi )$. It is proved that if $G$ is nilpotent, then there
exists a characteristic subgroup $C$ of nilpotency class bounded in
terms of $p$ such that the rank of $G/C$ is bounded in terms of $p$
and $r$.
For infinite (locally) nilpotent groups a similar result holds if the
group is torsion-free (due to Makarenko), or periodic, or finitely
generated; but examples show that these additional conditions cannot
be dropped, even for nilpotent groups.
As a corollary when $G$ is an arbitrary finite group, the
combination with the recent theorems of the author and Mazurov
gives characteristic subgroups $R\leqslant N\leqslant G$ such that
$N/R$ is nilpotent of class bounded in terms of $p$, while the
ranks of $R$ and $G/N$ are bounded in terms of $p$ and $r$ (under
the additional unavoidable assumption that $p\nmid |G|$ if $G$ is
insoluble); in general it is impossible to get rid of the
subgroup~$R$. The inverse limit argument yields corresponding
consequences for locally finite groups.